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Analytical solution of a new class of integral equations. (English) Zbl 1034.45001

The author gives an algorithm for finding analytically the unique solution \(h\in H^{-\alpha}(0,L)\) to the following equation \(Rh= f\), where \(Rh= \int^L_0 R(x,y) h(y)\,dy\), and the kernel \(R(x,y)\) satisfies the equation \(QR= P\delta(x-y)\), \(0\leq x\leq L\).
The above equation is the basic equation of random processes estimation theory. Finally, generalizations of the above equation in the multidimensional case are also considered.

MSC:

45A05 Linear integral equations
45H05 Integral equations with miscellaneous special kernels
60G60 Random fields
93E10 Estimation and detection in stochastic control theory